One-loop correction to the enhanced curvature perturbation with local-type non-Gaussianity for the formation of primordial black holes
De-Shuang Meng, Chen Yuan, Qing-Guo Huang
Abstract
As one of the promising candidates of cold dark matter, primordial black holes (PBHs) were formed due to the collapse of overdense regions generated by the enhanced curvature perturbations during the radiation-dominated era. The enhanced curvature perturbations are expected to be non-Gaussian in some relevant inflation models, and hence, the higher-order loop corrections to the curvature power spectrum might be non-negligible as well as altering the abundance of PBHs. In this paper, we calculate the one-loop correction to the curvature power spectrum with local-type non-Gaussianities characterized by ${F}_{\mathrm{NL}}$ and ${G}_{\mathrm{NL}}$ standing for the quadratic and cubic non-Gaussian parameters, respectively. Requiring that the one-loop correction be subdominant, we find a perturbativity condition, namely, $|2cA{F}_{\mathrm{NL}}^{2}+6A{G}_{\mathrm{NL}}|\ensuremath{\ll}1$, where $c$ is a constant coefficient which can be explicitly calculated in the given model, and $A$ denotes the variance of the Gaussian part of the enhanced curvature perturbation, and such a perturbativity condition can provide a stringent constraint on the relevant inflation models for the formation of PBHs.